1. Field of the Invention
The present invention relates to the technology of performing an electromagnetic field analysis based on the FDTD (Finite Difference Time Domain) method by dividing an analyzed space into hexahedral cells.
2. Description of the Related Art
The electromagnetic field analysis is performed in various fields on the antenna problem, the scattering of electromagnetic waves, etc. As analyzing methods, the FEM (Finite Element Method), the moment method, the BEM (Boundary element method), etc. have been used. Recently, the FDTD (Finite Difference Time Domain) method, which is an FDM (Finite Difference Method) suggested by K. S. Yee, has attracted much attention because of a number of advantages such as its simplicity in algorithm, high precision, wide application, etc.
In the FDTD method, an electromagnetic field is differentiated in a time domain and an analyzed space, and the behavior of an electromagnetic field is obtained by a numerical analysis based on the geometric arrangement and shape, and the electric (physical) characteristics such as the conductivity, magnetic permeability, etc. of an object to be analyzed. In a time domain, a discrete transformation is performed in a very short time step (xcex94t). An analyzed space is divided into cells for the discrete transformation.
FIG. 1 is an explanatory view of the cell dividing an analyzed space. As shown in FIG. 1, the cell is set as hexahedrons (normally cubes) each having the lengths of xcex94x in the x-axis direction, xcex94y in the y-axis direction, and xcex94z in the z-axis direction. The magnetic field intensity is placed in the center of each plane, and the field intensity is placed in the center of each side. Thus, the field rotation produces a magnetic field, and the magnetic field rotation produces an electric field by differentiating the magnetic field intensity from the field intensity by half-cell spatially shifting them from each other, thereby directly solving the Maxwell""s equation. The cell size is set to be less than {fraction (1/10)} of the shortest wavelength normally defined as a problem.
The electric field and the magnetic field differentiated by spatially shifted from each other are alternately arranged in a time domain as shown in FIG. 2. That is, the leap-frog algorithm for alternately computing the field intensity and the magnetic field intensity is used. Thus, the magnetic field intensity is obtained from the field intensity, and the field intensity is obtained from the magnetic field intensity, and these processes are alternately performed. The superscripts such as xe2x80x98nxe2x88x92xc2xdxe2x80x99, xe2x80x98nxe2x80x99, etc. shown in FIG. 2 represent the positions arranged in the time domain. The difference of the time domain is obtained such that the Courant stability condition can be satisfied.
In the FDTD method, the central difference method is used in differentiating the time domain and the analyzed space using the above mentioned leap-frog algorithm. Depending on the field in which an analysis is performed, the arrangements of the electric field and the magnetic field in a time axis or spatial coordinates can be inverse to each other.
The FDTD method is based on the Maxwell""s equation, which is also based on the Ampere""s circuital law, and the Faraday""s law of induction. Each law is represented in the differentiation form and the integral form, and the above mentioned Maxwell""s equation refers to a total of the four equations obtained by combining with the Gauss"" law on electric field and magnetic field. The Maxwell""s equation in the differentiation form is represented as follows using the field intensity E [V/m], the magnetic field intensity H [A/m], the electric flux density D [C/m2], the magnetic flux density B [T], the electric charge density xcfx81 [C/m3], and the current density J [A/m2].                               rot          ⁢                      xe2x80x83                    ⁢                      H            ⁡                          (                              r                ,                t                            )                                      =                                            ∂                              D                ⁡                                  (                                      r                    ,                    t                                    )                                                                    ∂              t                                +                      J            ⁡                          (                              r                ,                t                            )                                                          (        1        )                                          rot          ⁢                      xe2x80x83                    ⁢                      E            ⁡                          (                              r                ,                t                            )                                      =                              ∂                          B              ⁡                              (                                  r                  ,                  t                                )                                                          ∂            t                                              (        2        )                                          rot          ⁢                      xe2x80x83                    ⁢                      B            ⁡                          (                              r                ,                t                            )                                      =        0                            (        3        )                                          rot          ⁢                      xe2x80x83                    ⁢                      D            ⁡                          (                              r                ,                t                            )                                      =                  ρ          ⁡                      (                          r              ,              t                        )                                              (        4        )            
where the equation (1) (Ampere""s law), the equation (2) (Faraday""s law of induction), the equation (3) (Gauss"" law in the magnetic field), and the equation (4) (Gauss"" law in the electric field) are not independent equations. That is, the equations (3) and (4) are at most used as the evaluation references of a numerical error in the FDTD method, and the equations (1) and (2) are used in formulation. If the formulation is performed by applying the central difference method to the time domain and the spatial coordinate using the algorithm of Yee in the equations (1) and (2), then the following FDTD basic equation can be obtained in a 3-dimensional space.                                           E            x            n                    ⁡                      (                                          i                +                                  1                  2                                            ,              j              ,              k                        )                          =                                                            K                                  1                  ⁢                  x                                            ⁡                              (                                                      i                    +                                          1                      2                                                        ,                  j                  ,                  k                                )                                      ⁢                                          E                x                                  n                  -                  1                                            ⁡                              (                                                      i                    +                                          1                      2                                                        ,                  j                  ,                  k                                )                                              +                      
                    ⁢                      xe2x80x83                    ⁢                                                    K                                  2                  ⁢                  x                                            ⁡                              (                                                      i                    +                                          1                      2                                                        ,                  j                  ,                  k                                )                                      xc3x97                          {                                                                    [                                                                                            H                          z                                                      n                            -                                                          1                              2                                                                                                      ⁢                                                  (                                                                                    i                              +                                                              1                                2                                                                                      ,                                                          j                              ⁢                                                              xe2x80x83                                                            +                                                              1                                2                                                                                      ,                            k                                                    )                                                                    -                                              
                                            ⁢                                              xe2x80x83                                            ⁢                                                                        H                          z                                                      n                            -                                                          1                              2                                                                                                      ⁡                                                  (                                                                                    i                              +                                                              1                                2                                                                                      ,                                                          j                              ⁢                                                              xe2x80x83                                                            -                                                              1                                2                                                                                      ,                            k                                                    )                                                                                      ]                                    ⁢                                      xe2x80x83                                    ⁢                  Δ                  ⁢                                      xe2x80x83                                    ⁢                  z                                -                                  
                                ⁢                                  xe2x80x83                                ⁢                                                      [                                                                                            H                          y                                                      n                            -                                                          1                              2                                                                                                      ⁡                                                  (                                                                                    i                              +                                                              1                                2                                                                                      ,                            j                            ⁢                                                          xe2x80x83                                                        ,                                                          k                              +                                                              1                                2                                                                                                              )                                                                    -                                                                        H                          y                                                      n                            -                                                          1                              2                                                                                                      ⁡                                                  (                                                                                    i                              +                                                              1                                2                                                                                      ,                            j                            ⁢                                                          xe2x80x83                                                        ,                                                          k                              -                                                              1                                2                                                                                                              )                                                                                      ]                                    ⁢                                      xe2x80x83                                    ⁢                  Δ                  ⁢                                      xe2x80x83                                    ⁢                  y                                            }                                                          (        5        )                                                                    H              x                              n                +                                  1                  2                                                      ⁡                          (                              i                ,                                  j                  +                                      1                    2                                                  ,                                  k                  ⁢                                      xe2x80x83                                    +                                      1                    2                                                              )                                =                                                    H                x                                  n                  -                                      1                    2                                                              ⁡                              (                                  i                  ,                                      j                    +                                          1                      2                                                        ,                                      k                    ⁢                                          xe2x80x83                                        +                                          1                      2                                                                      )                                      -                          
                        ⁢                          xe2x80x83                        ⁢                                                            K                                      3                    ⁢                    x                                                  ⁡                                  (                                      i                    ,                                          j                      +                                              1                        2                                                              ,                                          k                      +                                              1                        2                                                                              )                                            ⁢                              xe2x80x83                            ⁢                              {                                                                            [                                                                                                    E                            z                            n                                                    ⁢                                                      (                                                          i                              ,                                                              j                                +                                1                                                            ,                                                              k                                ⁢                                                                  xe2x80x83                                                                +                                                                  1                                  2                                                                                                                      )                                                                          -                                                  
                                                ⁢                                                  xe2x80x83                                                ⁢                                                                              E                            z                            n                                                    ⁡                                                      (                                                          i                              ,                              j                              ,                                                              k                                ⁢                                                                  xe2x80x83                                                                +                                                                  1                                  2                                                                                                                      )                                                                                              ]                                        ⁢                    Δ                    ⁢                                          xe2x80x83                                        ⁢                    z                                    -                                                            [                                                                                                    E                            y                            n                                                    ⁢                                                      (                                                          i                              ,                                                              j                                ⁢                                                                  xe2x80x83                                                                +                                                                  1                                  2                                                                                            ,                                                              k                                +                                1                                                                                      )                                                                          -                                                                              E                            y                            n                                                    ⁡                                                      (                                                          i                              ,                                                              j                                ⁢                                                                  xe2x80x83                                                                +                                                                  1                                  2                                                                                            ,                              k                                                        )                                                                                              ]                                        ⁢                    Δ                    ⁢                                          xe2x80x83                                        ⁢                    y                                                  }                                                    ⁢                  
                                    (        6        )                                                      K                          1              ⁢              x                                ⁡                      (                                          i                +                                  1                  2                                            ,              j              ,              k                        )                          =                              1            -                                                            σ                  ⁡                                      (                                                                  i                        +                                                  1                          2                                                                    ,                      j                      ,                      k                                        )                                                  ⁢                Δ                ⁢                                  xe2x80x83                                ⁢                t                                            2                ⁢                                  xe2x80x83                                ⁢                                  ϵ                  ⁡                                      (                                                                  i                        +                                                  1                          2                                                                    ,                      j                      ,                      k                                        )                                                                                            1            +                                                            σ                  ⁡                                      (                                                                  i                        +                                                  1                          2                                                                    ,                      j                      ,                      k                                        )                                                  ⁢                Δ                ⁢                                  xe2x80x83                                ⁢                t                                            2                ⁢                                  xe2x80x83                                ⁢                                  ϵ                  ⁡                                      (                                                                  i                        +                                                  1                          2                                                                    ,                      j                      ,                      k                                        )                                                                                                          (        7        )                                                      K                          2              ⁢              x                                ⁡                      (                                          i                +                                  1                  2                                            ,              j              ,              k                        )                          =                                                            Δ                ⁢                                  xe2x80x83                                ⁢                t                                                              xe2x80x83                                ⁢                                  ϵ                  ⁡                                      (                                                                  i                        +                                                  1                          2                                                                    ,                      j                      ,                      k                                        )                                                                                      1              +                                                                    σ                    ⁡                                          (                                                                        i                          +                                                      1                            2                                                                          ,                        j                        ,                        k                                            )                                                        ⁢                  Δ                  ⁢                                      xe2x80x83                                    ⁢                  t                                                  2                  ⁢                                      xe2x80x83                                    ⁢                                      ϵ                    ⁡                                          (                                                                        i                          +                                                      1                            2                                                                          ,                        j                        ,                        k                                            )                                                                                                    ⁢                      1                          Δ              ⁢                              xe2x80x83                            ⁢              y              ⁢                              xe2x80x83                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢              z                                                          (        8        )                                                      K                          3              ⁢              x                                ⁡                      (                          i              ,                              j                +                                  1                  2                                            ,                              k                +                                  1                  2                                                      )                          =                                            Δ              ⁢                              xe2x80x83                            ⁢              t                                      μ              ⁡                              (                                  i                  ,                                      j                    +                                          1                      2                                                        ,                                      k                    +                                          1                      2                                                                      )                                              ⁢                      1                          Δ              ⁢                              xe2x80x83                            ⁢              y              ⁢                              xe2x80x83                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢              z                                                          (        9        )            
where xcexc, xcex5, and "sgr" respectively indicate the magnetic permeability, permittivity, and electric resistivity.
The characters xe2x80x98xxe2x80x99 and xe2x80x98yxe2x80x99 added as the subscripts to the respective symbols E and H in the equations (5) and (6) indicate the direction of the field intensity and the magnetic field intensity. In these equations, only the field intensity and the magnetic field intensity assigned in the x direction are shown, but they can also be derived in the y and z directions. The documents of the conventional technology to be referred to in understanding the present invention are: xe2x80x98FDTD (Finite Difference Time Domain) methodxe2x80x99 by Hano, 2002 National Convention Record IEE Japan, vol. 5, pp 411-414, 2002 (hereinafter referred to as the conventional technology document 1); xe2x80x98Development of a General Surge Analysis Program Based on the FDTD Methodxe2x80x99 by Noda and Yokoyama, The Transactions of the Institute of Electrical Engineers of Japan, vol. 121-B, No. 5, pp 625-632, 2001 (hereinafter referred to as the conventional technology document 2); xe2x80x98A Locally Conformed Finite-Difference Time-Domain Algorithm of Modeling Arbitrary Shape Planar Metal Stripsxe2x80x99 by J. Fang and J. Ren; IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 5, MAY 1993 (hereinafter referred to as the conventional technology document 3), etc.
The FDTD method is based on the division of an analyzed space by cells as shown in FIG. 1. Therefore, when an object to be analyzed such as a conductor which does not exist in the grid line (side) of the cell, etc. is represented, the shape of the object approximates to the shape of a staircase in the staircase approximation method. However, the shape represented in the staircase approximation method is normally quite different from the actual shape. Therefore, in the conventional analyzing method using the staircase approximation method, there has been the problem that there occurs a large error in the resonance frequency of an object to be analyzed and its propagation time, thereby failing in carrying out an analysis with high precision.
The difference between the shape represented in the staircase approximation method and the actual shape can be reduced by reducing the cell size. However, smaller cells require a larger number of cells, thereby increasing necessary computer resources and requiring a longer computing time. As a result, it is not desired to reduce the cell size even partially. Since a relatively large analyzed space can be processed in the field of a surge analysis, it is particularly not desired.
When the shape of an object to be analyzed is represented by transforming the shape of a cell, the modeling is required depending on the object, thereby lacking the generality. The method of setting the coordinate system such that the object can be in the grid lines normally cannot be used when a plurality of objects exist in an analyzed space at an arbitrary state. Thus, it is significant to more correctly represent and analyze an object to be analyzed existing in an analyzed space without reducing a cell size, transforming the shape of the cell, etc.
The present invention aims at providing a general-purpose technology for carrying out an electromagnetic field analysis with high precision and reliability by correctly representing an object to be analyzed (medium) existing in an analyzed space.
The electromagnetic field analysis method based on the FDTD method according to the present invention is a method of carrying out an electromagnetic field analysis using the FDTD (Finite Difference Time Domain) method by dividing an analyzed space into hexahedral cells. In this method, a medium selected from among a plurality of media existing in an analyzed space to be divided into cells is set as a line, and a plane in which the line crosses a plurality of sides in the planes forming the cell is divided by the line into plural units, thereby obtaining field intensity or magnetic field intensity.
In the above mentioned division of the plane by the line, it is desired that a section not containing the position in which the magnetic field intensity is computed in a plurality of sections obtained by dividing the plane by the line is included as an adjacent plane, and a section containing the position is not included as an adjacent plane. It is also desired that at least one of the field intensity of the plurality of the sections obtained by dividing the plane by the line as the field intensity having the same direction computed on the periphery of the plane. It is desired that the magnetic field intensity of the plane divided into a plurality of sections by the line is computed using the field intensity obtained using the integral loop including the line. It is also desired that the magnetic field intensity of each plane which is adjacent in the crossing direction to the plane divided by a line into a plurality of sections and has a side crossing the line is computed based on a plurality of adjacent sections. It is desired that each field intensity of the plane divided by a line into a plurality of sections is computed without considering the line, and then one or more of the computed field intensity is to be amended as necessary.
The medium representing method in the electromagnetic field analysis according to the present invention is a method used to represent a medium by the electromagnetic field analysis carried out in the FDTD method by dividing an analyzed space into hexahedron cells. In this method, a medium existing in an analyzed space to be divided into cells is selected, and the selected medium is set and represented as a line crossing a plurality of sides of a plane forming part of a cell.
The simulation device according to the present invention is based on the assumption that an analyzed space is divided into hexahedron cells, and an electromagnetic field analysis is carried out in the FDTD method, and includes a medium setting unit for setting a medium existing in the analyzed space as a line, and an analysis unit for obtaining the field intensity or the magnetic field intensity by dividing a plane crossing a plurality of sides in the planes of the cell obtained by dividing the analyzed space by the line set by the medium setting unit.
In the above mentioned division of the plane by the line, it is desired that a section not containing the position in which the magnetic field intensity is computed in a plurality of sections obtained by dividing the plane by the line is included as an adjacent plane, and a section containing the position is not included as an adjacent plane. In the analysis unit, it is also desired that at least one of the field intensity of the plurality of the sections obtained by dividing the plane by the line as the field intensity having the same direction computed on the periphery of the plane.
The storage medium according to the present invention is computer-readable, and stores a program executed by a computer to realize the simulation device according to the present invention.
According to the present invention, a medium (a conductor, a structure, etc.) selected from the media existing in an analyzed space is set (represented) as a line, and at least one of the field intensity and the magnetic field intensity is obtained after dividing a plane having a plurality of sides crossed by the line in the planes forming a cell.
By obtaining at least one of the field intensity and the magnetic field intensity after dividing the plane by the line into a plurality of sections, the line can be practically processed as arranged on the side of the plane of a cell. Since the line can be arranged on the plane of a cell, high flexibility can be obtained, thereby improving the generality. The medium represented by the line can produce imitation more correctly in shape. Thus, the electromagnetic field analysis can be performed with high precision and generality.
When, in the above mentioned division of the plane by the line, a section not containing the position in which the magnetic field intensity is computed in a plurality of sections obtained by dividing the plane by the line is included as an adjacent plane, and a section containing the position is not included as an adjacent plane, the increase of the number of planes produced by the division can be avoided. As a result, the number of the values of the finally obtained intensity, that is, the amount of data to be stored can be avoided, thereby suppressing the increase of necessary computer resources.